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**in the Loop**

### Stevine Obura Onyango

**To cite this version:**

Stevine Obura Onyango. Behaviour Modelling and System Control with Human in the Loop. Signal and Image Processing. Université Paris-Est, 2017. English. �NNT : 2017PESC1162�. �tel-01762668�

### WITH HUMAN IN THE LOOP

### by

### Onyango, Stevine Obura

### Submitted in partial fulfillment of the requirement for the degree

### DOCTORAT SIGNAL, IMAGE, AUTOMATIQUE

### in the

### UNIVERSIT´ E PARIS-EST CR´ ETEIL ET MSTIC Co-Tutelle

### TSHWANE UNIVERSITY OF TECHNOLOGY

### Pr´ esident du jury: Prof. Eric MONACELLI Directeurs de th´ ese: Prof. Boubaker DAACHI Directeurs de th´ ese: Prof. Yskandar HAMAM Directeurs de th´ ese: Prof. Karim DJOUANI

### Rapporteur: Prof. Hichem ARIOUI Rapporteur: Prof. Abdelouahab ZAATRI

### Examinateur: Prof. Anna PAPPA

### February 2017

I hereby declare that the dissertation submitted for the degree Doctor of Technology:

Electrical Engineering, at Tshwane University of Technology, is my own original work and has not previously been submitted to any other institution of higher education. I further declare that all the sources cited and quoted are indicated and acknowledged by means of a comprehensive list of references

Onyango S.O. Date: 24-February-2017

Power. Then will our world know the blessings of peace”

WILLIAM EWART GLADSTONE

Malgr´e le progr`es en recherche et d´eveloppement dans le domaine de systme au- tonome, de tels syst`emes n´ecessitent l’intervention humaine pour r´esoudre les probl`emes impr´evus durant l’ex´ecution des tˆaches par l’utilisateur. Il est donc n´ecessaire, malgr´e cette autonomie, de tenir compte du comportement du conducteur et il est difficile d’ignorer l’effet de l’intervention humaine dans le cadre de l’´evolution continue de l’environnement et des pr´ef´erences de l’utilisateur. Afin d’ex´ecuter les op´erations selon les attentes de l’operateur, il est n´ecessaire d’incorporer dans la commande les besoins de l’utilisateur. Dans les travaux pr´esent´es dans cette th`ese un mod`ele com- portemental de l’utilisateur est d´evelopp´e et int´egr´e dans la boucle de commande afin d’adapter la commande l’utilisateur. Ceci est appliqu´e la commande des fauteuils

´

electrique et assiste dans la navigation du fauteuil dans un milieu encombr´e. Le d´eveloppement du mod`ele comportemental est bas´e sur la m´ethode de potentielles orient´ees et la d´etection des obstacles et le comportement du conducteur vs de ces obstacles par l’adaptation du. L’´etude contribue ´egalement au d´eveloppement d’un mod`ele dynamique du fauteuil utilisable dans des situations normales et exception- nelles telle que le d´erapage. Ce mod`ele est d´evelopp´e pour un le cas le plus courant des fauteuil avec roues arri`ere conductrices utilisant le formalisme Euler Lagrange avec les forces gravitationnelles et sur des surfaces inclin´ees. Dans la formulation de la commande, le mod`ele du conducteur est introduit dans la boucle de com- mande. L’optimalit´e de la performance est assur´ee par l’utilisation du command pr´edictif g´en´eralis´e pour le syst`eme en temps continue. Les r´esultats de la simulation d´emontrent l’efficacit´e de l’approche propos´ee pour l’adaptation de la commande au comportement du conducteur.

Although the progressive research and development of autonomous systems is fairly evident, such systems still require human interventions to solve the unforeseen com- plexities, and clear the uncertainties encountered in the execution of user-tasks. Thus, in spite of the system’s autonomy, it may not be possible to absolutely disregard the operator’s role. Human intervention, particularly in the control of auto-mobiles, may as well be hard to ignore because of the constantly changing operational context and the evolving nature of the drivers’ needs and preferences. In order to execute the autonomous operations in conformity with the operator’s expectations, it may be necessary to incorporate the advancing needs and behaviour of the operator in the design. This thesis formulates an operator behaviour model, and integrates the model in the control loop to adapt the functionality of a human-machine system to the op- erator’s behaviour. The study focuses on a powered wheelchair, and contributes to the advancement of steering performance, through background assistance by mod- elling, empirical estimation and incorporation of the driver’s steering behaviour into the control system. The formulation of the steering behaviour model is based on two fundamentals: the general empirical knowledge of wheelchair steering, and the steer- ing data generated from the virtual worlds of an augmented wheelchair platform. The study considers a reactive directed potential field (DPF) method in the modelling of drivers’ risk detection and avoidance behaviour, and applies the ordinary least square procedure in the identification of best-fitting driver parameters. The study also con- tributes to the development of a dynamic model of the wheelchair, usable under normal and non-normal conditions, by taking into consideration the conventional differential drive wheelchair structure with two front castor wheels. Derivation of the dynamic model, based on the Euler Lagrange formalism, is carried out in two folds: initially by considering the gravitational forces subjected to the wheelchair

slipping parameters into the model. Determination of the slipping parameters is ap- proached from the geometric perspective, by considering the non-holonomic motions of the wheelchair in the Euclidean space. In the closed-loop model, the input-output feedback controller is proposed for the tracking of user inputs by torque compen- sation. The optimality of the resulting minimum-phase closed-loop system is then ensured through the performance index of the non-linear continuous-time generalised predictive control (GPC). Simulation results demonstrate the expected behaviour of the wheelchair dynamic model, the steering behaviour model and the assistive capability of the closed-loop system.

I sincerely express my gratitude and appreciation to Prof. Yskandar Hamam, who apart from being my academic study leader has also been a father and a role model.

I express my special appreciation to Prof. Karim Djouani and Prof Baubaker Daachi whose scientific contributions and academic guidance were indispensable throughout the entire duration of this research. I also can’t forget Dr. Nico Steyn who worked tirelessly with me, and was always available for me.

A special ‘thank you’ to The National Research Fund (NRF) and The South African government and People at large, to F’SATI and Electrical Engineering Department at TUT, for granting me the opportunity to conduct my research and the basic framework from which my studies were possible, I thank you.

To my brothers Kenns and Elisha, my sister Millicent, my father Benson and my mother Fellidah, and most of all to my wife Eglen, your untiring support and en- couragement in all matters made a big difference. May the Almighty God bless you according to His will.

Dr. Maina Mambo, Dr Michael Ajayi and Dr. Amos Anele thank you for always lending me a patient ear whenever I posed my numerous questions.

Declaration and Copyright i

Abstract iii

Acknowledgements vi

List of Figures xi

List of Tables xiv

Abbreviations xv

Symbols xvi

1 INTRODUCTION 1

1.1 Background and motivation of the study . . . 3

1.2 Problem statement . . . 6

1.2.1 Sub-problem 1 . . . 6

1.2.2 Sub-problem 2 . . . 7

1.2.3 Sub-problem 3 . . . 7

1.3 Research Objectives . . . 8

1.4 Methodology . . . 9

1.5 Outline of the main contributions . . . 11

1.6 Delineations and Limitations . . . 12

1.7 Publications . . . 13

1.8 Thesis Chapter Overview . . . 13

2 CONTROL WITH HUMAN-IN-THE-LOOP METHODOLOGIES: A SURVEY 15 2.1 Introduction . . . 15

2.2 Background . . . 15

2.3 Modelling of WMSs . . . 17

2.3.1 Special kinematic characteristics of a WMS . . . 18

2.3.2 Kinematic modelling of a differential drive system . . . 20

2.3.2.1 Coordinate system assignment . . . 20

2.3.2.2 The homogeneous transformation matrix . . . 22

2.3.2.3 Forward and Inverse Kinematic Solutions . . . 24

2.3.2.4 The kinematic model of a differential drive system: An example . . . 25

2.3.3 Dynamic Modelling of differential drive systems . . . 26

2.3.3.1 The Newton-Euler formulation . . . 27

2.3.3.2 The Euler-Lagrange dynamics . . . 28

2.3.3.3 Newton-Euler versus Euler-Lagrange . . . 29 vii

2.4 Operator behaviour modelling . . . 32

2.4.1 Existing driver behaviour models . . . 33

2.4.1.1 The control theoretic models . . . 34

2.4.1.2 The information processing models . . . 35

2.4.1.3 The motivational models . . . 36

2.4.1.4 Hierarchical models . . . 38

2.4.2 Driver behaviour models for path and speed planning . . . 39

2.4.3 The context around the use of wheelchairs . . . 40

2.4.4 Existing wheelchair driver and steering models . . . 41

2.5 System control with human-in-the-loop . . . 43

2.5.1 Shared control in general applications . . . 44

2.5.2 Application in motor-vehicle and wheelchair control . . . 46

2.5.3 Control theoretical tools . . . 50

2.5.3.1 Model predictive control . . . 51

2.5.3.2 Feedback linearisation . . . 53

2.6 Feedback linearisation procedure . . . 55

2.6.1 Zero Dynamics . . . 57

2.7 Conclusions . . . 58

3 MODELLING A POWERED WHEELCHAIR WITH SLIPPING AND GRAVITATIONAL DISTURBANCES 59 3.1 Introduction . . . 59

3.2 Background and Motivation . . . 60

3.3 Dynamic model with gravitational forces . . . 63

3.3.1 Description of the wheelchair and frames of reference . . . 63

3.3.2 System constraints . . . 64

3.3.3 Kinetic and potential energy . . . 66

3.3.4 Dynamic model development . . . 68

3.4 Slipping parameters and frictional force . . . 69

3.4.1 Slipping parameters . . . 69

3.4.2 Determination of real velocity . . . 71

3.4.3 Frictional and resistive force modelling . . . 75

3.5 Simulation and results . . . 77

3.5.1 A comparison: the model with and without rolling friction . . 78

3.5.2 A comparison: the model with and without gravitation effects 80 3.5.3 A comparison with other differential drive models . . . 84

3.5.4 Simulation with a slipping disturbance . . . 84

3.6 Conclusions . . . 85

4 A DRIVING BEHAVIOUR MODEL OF ELECTRICAL WHEEL- CHAIR USERS 87 4.1 Introduction . . . 87

4.1.1 Background and Motivation . . . 87

4.2 Path planning and driver adaptation models . . . 89

4.2.1 The potential field method . . . 91

4.2.2 DPF and other modified APF methods: A comparison . . . . 93

4.3 Simulator evaluation and steering data . . . 96

4.3.1 Evaluation of the VS-1 simulator . . . 96

4.3.2 Steering data and implied behaviour . . . 100

4.3.2.1 A risk free environment . . . 101

4.3.2.2 A minimal risk environment . . . 102

4.3.2.3 A living-room environment . . . 103

4.4 Modelling the driving behaviour . . . 104

4.4.1 Dynamic representation of driving behaviour . . . 107

4.4.2 Desired steering velocity . . . 108

4.4.3 Influence of risk and driver adaptation mechanisms . . . 109

4.5 Simulation, results and discussion . . . 111

4.5.1 Parameter identification and adaptation mechanism . . . 111

4.5.2 Trajectory fitting . . . 115

4.5.3 A comparison with Emam et al (2010)’s driver behaviour model 118 4.6 Conclusion . . . 119

5 A CLOSED-LOOP CONTROL WITH HUMAN-IN-THE-LOOP 122 5.1 Introduction . . . 122

5.2 The control tool . . . 123

5.3 Feedback linearisation background . . . 124

5.4 Configuration of the control system . . . 126

5.5 System description . . . 127

5.6 Feedback linearisation and controller design . . . 129

5.6.1 Navigation task . . . 130

5.6.2 Relative degree of the system (ρ) . . . 131

5.6.3 The control law . . . 131

5.7 Non-linear continuous-time GPC . . . 132

5.7.1 Closed-loop stability of the wheelchair system . . . 135

5.8 Simulation results of the closed-loop model . . . 136

5.8.1 Simulation without the driving behaviour model . . . 136

5.8.2 Simulation with the driving behaviour model . . . 138

5.8.3 Simulation with induced disability . . . 140

5.9 Conclusions . . . 145

6 CONCLUSION AND FUTURE WORK 146 6.1 Conclusion . . . 146

6.2 Recommendations for future works . . . 149

Bibliography 151

A Derivation of Equations 3.17-3.19 179

2.1 A simple closed chain mechanism . . . 19 2.2 A pictorial description of the driver model presented by Kondo &

Ajimine (1968) . . . 34 2.3 Okuda et al. (2014)’s predictive driver assisting system in a single car 47 2.4 The Qinan Li et al. (2011)’s architecture of the dynamic shared control 50 2.5 The structural procedure of feedback control . . . 54 3.1 A differential drive wheelchair model. . . 63 3.2 Geometrical representation of the wheelchair, with the parameters that

have been utilised in deriving the velocity of the centre of mass from the castor wheels’ velocities. . . 72 3.3 (a) Bird view and (b) side view schematic representation of a castor

wheel. . . 73 3.4 Straight line trajectories of wheelchair’s centre of mass generated by

torques τ_{R} =τ_{L} on a flat surface from coordinate [0 0 0] in 20 seconds. 79
3.5 The velocity curves for trajectories (A) and (C) respectively in Figure

3.4. . . 79
3.6 Straight line trajectories and rates of change ofx_{g}, y_{g} andz_{g} generated

from an inclined surface form an initial wheelchair position of [0 0]. . 80 3.7 Circular wheelchair trajectories and rates of change of xg, yg and zg

generated on a flat surface from initial position [0 0] and initial direc-
tion φ = 0^{◦} with τ_{R} and τ_{L} equal to 4 and 3, and 3 and 4 in the first
and the second sub-plot respectively. . . 81
3.8 Trajectories and velocities generated when initial wheelchair orienta-

tion is neither directly up nor directly down the slope. The simulation
have been conducted on a surface inclined byθ= 15^{◦} andψ = 0^{◦} from
an initial [0 0] wheelchair position. . . 83
3.9 The trajectory observed due to ψ on a surface inclined by θ= 15^{◦} and

ψ = 90^{◦} from an initial [0 0] wheelchair position. . . 83
3.10 Resulting deviation from the intended trajectory on a flat surface with

slight slip introduced into the model. . . 85 4.1 Virtual-reality System 1 (VS-1): The augmented virtual and motion

simulator at FSATI for wheelchair simulations. . . 97 4.2 The roller system on the motion platform that enables both rotational

motion of the wheels, and the mapping of the wheel’s motion into the virtual world. . . 98 4.3 A user steering the wheelchair in a living room set-up in both virtual

and reality environments. . . 99 4.4 Virtual-reality System 1 (VS-1): The augmented virtual and motion

platform at FSATI for wheelchair simulations. . . 100 4.5 Wheelchair trajectories and speeds observed in a risk free environment. 102

xi

4.6 Trajectories and speeds of wheelchair observed in a minimal risk envi- ronment. . . 103 4.7 The considered virtual living room environment (perimeter wall not

shown for clarity reason). . . 104 4.8 Wheelchair trajectories observed in a living-room environment. The

rectangular shapes in the configuration space represent the living room furniture. . . 105 4.9 Wheelchair speeds observed in the living room environment. . . 106 4.10 Influence of risky situations on wheelchair steering with m and n =

2, and with distance-to-risk d_{risk} considered as the main adaptation
reference. . . 111
4.11 The generated linear velocity and model response in Case 1 and Case 7.116
4.12 Regression errors in Case 1 and Case 7. . . 117
4.13 Generated trajectories and linear velocities in Case 1. . . 117
4.14 Generated trajectories and linear velocities in Case 7. . . 118
4.15 The curve-fitting comparison between the presented model and Emam

et al. (2010)’s Model . . . 119 5.1 The control diagram of a wheelchair with integrated driver behaviour

model and intention detection model. . . 127 5.2 Circular wheelchair trajectory generated by considering a ramp input

for reference angular orientation and V_{r} = 1.5 at θ = 0^{◦} and ψ =
90^{◦}. As depicted in time series curve for wheel torques, random slip
introduced at time t = 20s for the rest of simulation time does not
affect the ability of controller to automatically adjust the torques in
order to track the specified user inputs. . . 138
5.3 Sinusoidal wheelchair trajectory generated by considering a sine wave

input for reference angular orientation and V_{r} = 1.5 at θ = 0^{◦} and
ψ = 90^{◦}. Similarly, the random slip introduced at time t = 20s does
not affect the ability of controller to regulate wheel torques. . . 139
5.4 The original trajectory generated from the speed and directional com-

mands of the driver and the new controller computed trajectory in Case 1 and Case 7. . . 140 5.5 The original and controller computed linear speeds and their corre-

sponding errors in Case 1 and Case 7. . . 141 5.6 The resulting effect of the disability model on the angular velocity and

linear acceleration signals and its corresponding contribution on the angular position and linear velocity. . . 142 5.7 The wheelchair trajectory of a normal driver, the trajectory with su-

perimposed steering disability and the resulting controller generated wheelchair trajectory, in Case 1 and Case 7. . . 143

5.8 Sub-plots A and B depict linear wheelchair speeds produced by a nor- mal user produced, a disabled user and the human-in-the-loop con- troller, while sub-plots C and D shows the resulting velocity errors in the disabled and controller computed signals relative to the normal user’s, in Case 1 and Case 7. . . 144 A.1 A magnification from FIGURE 3.2 . . . 179

3.1 The dynamic model parameters used in simulation . . . 78 3.2 A comparison of the presented wheelchair model with other wheelchair

models. . . 84 4.1 Comparison of the potential field modifications based on their appli-

cability in the formulation steering behaviour of wheelchair users. . . 95 4.2 Statistical analysis of the model employing DTR as the adaptation

mechanism. The indicated values of constant p represent those that resulted in the highest coefficient of determination. . . 114 4.3 Statistical analysis of the model employing TTR as the main adapta-

tion mechanism, with the same constants as Table 4.2. . . 115 4.4 The regression parameters obtained with Emam et al. (2010)’s model 120 5.1 Dynamic model and controller parameters used in simulation . . . 137

xiv

ADAS Advance Driver Assistance Systems APF Artificial Potential Field

BEA Bacterial Evolutionary Algorithm D-H Denavit-Hartenberg convention DoF Degrees of Freedom

DPF Directed Potential Field DTR Distance-to-risk

EAPF Evolutionary Artificial Potential Field GNRON Goals Non-reachable with Obstacles Nearby GPC Generalised Predictive Control

HMD Head Mounted Display IDM Intelligent Driver Model LOA Level of Autonomy MDP Markov Decision Process

MOEA Multi-Objective Evolutionary Algorithm MPC Model Predictive Control

NN Neural Network

PEAPF Parallel Evolutionary Artificial Potential Field POMDP Partially Observable Markov Decision Process RHC Receding Horizon Control

SPRT Sequential Probability Ratio Test TTR Time-to-risk

VSL Variable Speed Limit WMS Wheeled Mobile System

xv

B(·)_{A} −→ (·) is a variable of (frame) A in/ w.r.t (frame) B

A(·) −→ (·) is a variable in (frame) A
(·)^{T} −→ Transpose of variable (·)

(·)˙ −→ First derivative of variable (·) w.r.t time (·)¨ −→ Second derivative of variable (·) w.r.t time

C −→ Centre of mass

(·)_{g} −→ Coordinate components of pointC in{x y z}
(·)˙ _{f} −→ Castor wheel velocity component

O −→ Mid-point of rear axle/origin of the body fixed frame
(·)˙ _{o} −→ Velocity components ofO as translated from C
(·)˙˜_{g} −→ Component of slipping velocity

v_{O} −→ Velocity ofO as translated from the centre of front axle
(·)˙ _{f}

o −→ Components ofv_{O}

¯

x,y,¯ z¯ −→ Unit basis coordinates

¯¯

x,y,¯¯ z¯¯ −→ Cartesian components of distance l between C and O {x y z} −→ Inertial coordinate frame

x, y, z −→ Cartesian coordinates of{x y z}

{X Y Z} −→ Body fixed frame

X, Y, Z −→ Cartesian coordinates of{X Y Z}

I −→ Inertia tensor

I_{XX}, I_{Y Y}, I_{ZZ} −→ Moment of inertia about X,Y and Z axis through O
I_{XZ} −→ Product inertia aboutX and Z axis through O
I −→ Composite matrix of identity matrices

V −→ Linear speed

ν −→ Linear velocity

ν −→ Velocity vector

ν_{0X}, ν_{0Y}, ν_{0Z} −→ Components of inertial velocity ofC along the instantaneous
directions of X, Y and Z axis

v_{R},v_{L} −→ Linear velocities of the right and left castor wheels
xvi

v_{RA},v_{LA} −→ Components ofv_{R} and v_{L} about point A
v_{RB},v_{LB} −→ Component ofv_{R} and v_{L} about point B
V_{r} −→ Reference linear speed

X −→ Distance

r −→ Position

T −→ Homogeneous transformation matrix T −→ Kinetic energy w.r.t point C

J −→ Wheel Jacobian matrix

B_{0} −→ Block diagonal matrix of wheel Jacobian matrices

R −→ Transformation matrix of frame{X, Y, Z}relative to{x, y, z}

b −→ Half rear axle length

D −→ Decoupling matrix

D_{v} −→ Directivity factor

˙

γ −→ Rotational velocity vector of the driving wheels

˙

γ −→ Average rotational velocity of the driving wheels

˙

γ_{R} ,γ˙_{L} −→ Angular velocities of the right and left wheels

˙

γ_{f}_{R}, γ˙_{f}_{L} −→ Angular velocities of the right and left castor wheels
Γ˙ −→ Composite vector of wheel velocity vectors

N −→ Normal force at pointO

¯

e −→ Unit vector in the direction of motion

¯

e_{o} −→ Unit vector in the direction of moving obstacle
q −→ Generalised coordinates vector

q_{i} −→ Component of the generalised coordinate vector

˙

q_{s} −→ Generalised velocity vector with slipping velocities

˙

q_{} −→ The new generalised velocity vector that includes the slipping
parameters

Q −→ External forces aiding or resisting the motion r −→ Radius of the driving wheels

r_{C} −→ Radius of the castor wheels
R^{2} −→ Coefficient of determination

θ −→ Instantaneous angular deviation of theZ-axis from thez-axis ψ −→ The angle between the x-axis and the line of intersection of

the movingXY plane and the stationary xy plane

φ −→ Precession angle about the Z-axis in a counter-clockwise di- rection as visible in the body fixed frame (yaw angle)

φ_{r} −→ Reference angular position

L −→ Lagrangian function

U −→ Potential energy

U_{art} −→ APF function

U_{att} −→ Attractive potential
U_{rep} −→ Repulsive potential

u −→ Input/control vector

U −→ State feedback law

µ −→ Coefficient of rolling friction

µ_{vis}, µ_{cmb} −→ Coefficients of viscous friction and Coulomb friction
M −→ Mass of the wheelchair including all its components
M(q¯ _{i}) −→ Symmetric positive definite inertia matrix

G(q_{i}) −→ Vector of gravitational forces

C(q¯ _{i},q˙_{i}) −→ Matrix of centrifugal and Coriolis forces

A(q) −→ Matrix associated with constraints of the system

˙

α_{R} −→ Orientational velocity of the right castor wheel

˙

α_{L} −→ Orientational velocity of the left castor wheel

A −→ Adaptation mechanism

a_{k}, ω_{k} −→ Linear acceleration and angular velocity respectively
λ −→ Vector of the Lagrange multipliers

χ −→ State vector

Y −→ State space output vector

ρ −→ Relative degree

ρ_{r} −→ Local roadway curvature

L_{f}h(x) −→ Lie derivative of h(x) along f_{1}(x)

L_{g}h(x) −→ Lie derivative of h(x) along g_{1}(x)
z =ϕ(χ) −→ Non-linear transformation

F( ˙q) −→ Vector of frictional forces E(q) −→ Input transformation matrix E(t)˜ −→ Vector of errors

e −→ Linear speed and angular position errors

−→ Vector of slipping parameters

τ −→ Vector of input torque

τ_{R}, τ_{L} −→ Right and left rear wheel torques
τ_{r} −→ Relaxation/reaction time

¯

τ −→ Duration of prediction

ω −→ Angular velocity

ω_{X}, ω_{Y}, ω_{Z} −→ Components ofω along the X, Y and Z axis

S(q_{i}) −→ Full rank transformation matrix which transforms η to ˙q
s_{r}, s_{h}, s_{w} −→ Slip ratio, hand stiffness and wrist stiffness

t, k −→ Time and time instant
k_{ν}, k_{env} −→ Driving model parameter

h −→ Output vector

T_{1}, T_{2} −→ Minimum and maximum prediction time respectively
k_{(·)} m, n, p −→ Constants

w, B, K N −→ Constants

## and

## most importantly

## my best friend and wife Eglen, my daughter Fellidah,

## my son Jesse, and

## my sister Millicent.

xx

### INTRODUCTION

The recent robotic advancements and autonomous controls are quite evident in many areas of application. However, these have not relieved the human fully from the oper- ator role. Human still intervenes to clear the complexities and environmental uncer- tainties that machines encounter during the operation. As a result, operator support and assistive systems, like autopilots in aircraft and advance driver assistance systems (ADAS) in auto-mobiles, have been developed to optimise the operator interventions through warnings and use of automated systems in the control and management of service machines. In auto-mobiles for instance, the driver assistance systems con- stantly interact with the driver to ensure a faultless operation of the vehicle. Indeed, the correct operation of such systems depends not only on the automated assistive system, but also on the handling behaviour of the human operator.

It may be known in general, that different operators will exhibit diverse handling behaviours and preferences, when operating similar systems under similar conditions.

Besides, human operators appreciate the assistance not only because the assistive system can perform the intended task, but also, based on the way the support system executes the assistance. This complicates the ability to realise the assistance that the operator appreciates, without synthesising the operator’s handling behaviour into the control system. Although the adaptability of assistive systems to the operator’s handling behaviour may not be very necessary in some applications, the operator- specific assistive systems become very essential in applications where the service machine forms an integral component of the user’s life, like in the use wheelchairs.

One way of realising the operator-specific assistance is through modelling and incor- porating the handling behaviour of the operator in the control system. The modelling

1

of handling behaviours entails the formulation of structures, and specifications of pro- cesses, that simulates the control actions of the operator. It involves the definition of initial conditions and operational limits of the operator and machine, with parallel consideration of the prevailing operational context and presumed assumptions. This, however, does not guarantee the possibility of finding a formulation that absolutely replicates the contextual handling behaviour of a human operator. As a result, it is almost inevitable, that some elements of human control are necessary to realise proper operation of such assistive systems.

The operator handling/behaviour models, have been formulated in different fields for various reasons, including design and accident analysis. The modelling of handling behaviour for design reasons is concerned with the development and assessment of different machine procedures and interfaces; while the modelling for accident analysis regards the causes of events that result in the human behaviours, for the sake of investigations. Coincidentally, most behaviour models have supported the design and development of dynamic and assistive systems, regardless of the modelling reason. In motor-vehicle safety assessment for instance, the driver models formulated to assist in the understanding and planning of solutions to traffic bottlenecks (Ahmed, 1999), have also aided the design and development of ADAS (Panou et al., 2007).

In order to design and evaluate the assistive system that takes into consideration the operator’s handling behaviour, it may be necessary that the appropriate machine model and control architecture are available. Balanced equations derived from the Newton’s laws and the virtual work concept, are generally used to express a machine’s behaviour. Indeed, different procedures consisting of D’Alembert’s, Lagrange’s and Hamilton’s can be used to formulate the basic laws of dynamics by considering the states of physical variables and functional components of a dynamic system. In particular, this study focuses on the standard powered wheelchair, and contributes to

the advancement of steering performance through background assistance. It involves modelling, empirical estimation, and incorporation of a wheelchair model and the driver’s steering behaviour model into the control system.

### 1.1 Background and motivation of the study

Wheelchair prevalence could be linked to the role they play in alleviating mobility restrictions over short distances. According to the South African profile report of persons with disability (Statistics South Africa & Lehohla, 2014), 2.3% (≈1.2 million) of the total South African population (≈ 52 million) depend on the wheelchair.

Moreover, the percentage of people in need of wheelchairs could be much higher in other underdeveloped countries because diseases responsible for mobility impairments like cerebral palsy can be associated with lower socio-economic status (Sundrum et al., 2005). While this may seem to represent a marginal portion of the population, it may not be possible to over emphasise the important sense of independence and self-esteem, that users with debilitating impairments experience with wheelchairs. It may be noted in the absence of wheelchairs and other mobility aids, that ambulatory impairments may result in extreme emotional loss, neglect, stress and even isolation (Finlayson & van Denend, 2003).

Normally, manual or powered wheelchairs can be used by individuals with physical lower limb impairments. However, the manual wheelchairs present difficult physical demands for users with both physical and cognitive impairments. On the other hand, powered wheelchairs eliminate the physical demands, but necessitate special control skills that some potential users do not possess (Simpson et al., 2008). In order to accommodate such users, the contemporary wheelchair research is focused towards user-suited interfaces and autonomous control.

Several robotic functionalities with computers and sensors have been considered in the design of autonomous wheelchairs in order to provide the user with a variety of hands-free navigation capabilities. Nonetheless, the studies and developments that regard wheelchair control for the sake of driver assistance and rehabilitation are still limited in spite of the current advancements. According to Fehr et al. (2000), about 10% of the users still encounter considerable difficulties in their daily use, and upto 40% find the steering task close to impossible. Besides, Fehr et al. (2000) observe that some potential users with multiple sclerosis and high-level spinal cord injuries, have spent extremely long training and rehabilitation durations with insignificant success.

In the absence of caretakers, such individuals may not be able to use the wheelchair.

Encompassing this user group necessitates assistive improvements in the control and management of wheelchair systems.

In order to empower debilitated individuals with the full independence and self-esteem that the stronger users experience, it is important that the high-level decision making tasks and control process are granted to the user, and not the autonomous wheelchair controller. This means, that the user may still need to perform the ordinary steering manoeuvres with necessary assistance and a suitable interface. Assisting a driver who is in active control, may require the system to determine, in the appropriate way, the assistive adjustment as well as the extent to which the adjustment is provided. The control system therefore needs to be aware of the intention and steering preferences of the driver. It is considered that this could be realised by taking the driver’s steering behaviour into consideration in the design of the wheelchair’s control system. This is regarded as control with the user-in-the-loop.

According to Panou et al. (2007), drivers are known to adjust their speed in order to establish the equilibrium between the environmental situation and the acceptable subjective risk. This compensatory mechanism is proposed, to adapt the steering

control of the wheelchair to the driver’s handling behaviour. It is presumed that the approach reduces the driver’s workload in fine control, and provides steering assis- tance regardless of the impairment condition. Besides, the assistance could remedy the effects of functional deterioration and fatigue and improve the comfort and safety of the driver.

Validating the assistive system necessitates the formulation of a wheelchair model and proper implementation of a control architecture. It is important that the ac- tual behaviour of the wheelchair is represented as much as possible by the model.

In literature, the modelling of differential drive wheelchairs is carried out from both kinematics and dynamic perspectives. The kinematic models present ideal formula- tions that relate the wheel rates of the wheelchair to the body-fixed frame velocities, by considering the geometric properties. However, kinematic models do not account for the effects of mass, inertia and acceleration, and are therefore used with antic- ipation that the controller will be robust enough to account for the unconsidered dynamical properties (Tarokh & McDermott, 2005; Tian et al., 2009; Zhang et al., 2009).

Dynamic modelling on the other hand, incorporates both kinematic and dynamic properties of the system. For this reason, dynamic modelling is one of the common modelling approaches in the literature. Most dynamic models, however, presume two dimensional configurations with pure rolling constraints, and rarely account for the combined effects of wheel slip, frictional resistance and gravitational disturbance (Oubbati et al., 2005; Koz lowski & Pazderski, 2004). As a result, such models may not be comprehensive enough to reflect the actual outdoors behaviour of the wheel- chair. It is suggested, that one solution to wheelchair automation and performance improvement is through better and realistic system modelling.

### 1.2 Problem statement

A large percentage of the wheelchair user community can steer with confidence in adequate environments. However, the steering accuracy varies with the kind of im- pairment, the extent of disability and the inherent monotony of wheelchair steering (Fehr et al., 2000). Extreme cases of the steering inaccuracies have caused collision accidents in typical residential settings (Fehr et al., 2000; Cooper et al., 1996; Rodgers et al., 1994). Besides, the available wheelchair control techniques have not addressed the progressive deterioration in the steering capability of users with degenerative con- ditions (Ando & Ueda, 2000). Accordingly, this thesis intends not only to address the above steering inaccuracy problem, but also seeks to ensure that the desired accuracy and the ensuing steering assistance is adapted to the driver’s steering behaviour, and benefits the whole user community regardless of the disability condition. It is consid- ered that this could be achieved through better modelling and system control. There is need, therefore, that an appropriate control architecture and a realistic model of the wheelchair and the driver’s steering behaviour is available. Thus, the following sub-problems are observed:

### 1.2.1 Sub-problem 1

Wheelchair driving or driver models are considerably few in the literature. Besides, the available models, suffer lack of individuality, focusing mostly on common user attributes, and assume that all drivers respond to navigation situations by similar general patterns (Diehm et al., 2013). Such driver models employ the general param- eters that barely correspond to measurements obtained from extreme drivers, and hardly take into consideration the contextual nature of human response to stimuli.

It is therefore important that a wheelchair steering model, capable of addressing the specific steering behaviour of the driver, is formulated.

### 1.2.2 Sub-problem 2

The disability condition of the wheelchair user-community requires assistive systems that take into consideration the favourable indoor as well as the unstructured out- door steering situations. The available wheelchair models, however, fail to take into consideration the aggregated effects of extreme dynamic steering situations on the wheelchair (Zhu et al., 2006; Tian et al., 2009; Zhang et al., 2009). This therefore necessitates the formulation of a comprehensive wheelchair model that presumes the unstructured and structured dynamic conditions, and takes the effects of gravitational forces, wheel-slip and rolling friction on the usable-traction into consideration.

### 1.2.3 Sub-problem 3

The existing steering assistance solutions provide discrete levels of shared control;

with full computer control at the autonomous level, and full driver control at the operator level (Rofer & Lankenau, 1999; Levine et al., 1999). The driver is tasked with the responsibility of choosing the appropriate control level, and the intended destination in the case of full autonomous control. It is considered, that choosing the destination and control mode may constitute a cognitively challenging responsibility to some users. Moreover, a particular path to the destination may be preferred;

if the wheelchair is steered autonomously to the goal without necessarily following the preferred path, the driver may fail to appreciate the assistance. It is therefore important that the control and decision making tasks are granted the driver, while the steering assistance is executed by the steering behaviour model in the control

loop. In this regard, a pertinent control architecture is required to accomplish the steering assistance.

### 1.3 Research Objectives

The primary objective of this study is to advance the state of the art in wheelchair steering, by synthesising the driver’s handling behaviour into the control system, so as to provide a driver-specific background steering assistance. This involves integrat- ing the wheelchair dynamic model and the driving behaviour model of the user in the control system, to adapt the control of the wheelchair to the driver’s steering behaviour.

The following objectives are therefore considered:

• To formulate and verify a versatile empirical driving model of a powered wheel- chair user, based on the observable actions of the driver, with particular con- sideration of the steering signals and the prevailing environmental situation.

• To identify the steering behaviour of the driver in terms of the driving model’s parameters, using the steering data generated from the virtual worlds of an augmented wheelchair platform.

• To formulate and validate a dynamic model of a differential drive powered wheelchair, that takes into consideration the effects of rolling friction and grav- itational potential of the wheelchair, on both inclined and non-inclined surfaces.

• To implement a control system that incorporates the wheelchair model and the driving behaviour model in the control loop, in order to adapt the steering of the wheelchair to the driver’s behaviour and realise the intended steering support.

### 1.4 Methodology

Human-in-the-loop control is increasingly becoming one of the acceptable concepts of realising the provisional demands of semi-autonomous controllers that occasionally necessitate human intervention (Rothrock & Narayanan, 2011; Chiang et al., 2010;

Tsui et al., 2011; Stoelen et al., 2010; Smith, 2003). Human-in-the-loop approach to system control is adopted and implemented in this study using the classical feedback control technique (Isidori, 1995; DeFigueiredo & Chen, 1993). The idea is executed by synthesising the wheelchair model and the discrete reactive model of the driver’s steering behaviour in the control system. The assistive control with human-in-the- loop is effected in three stages.

At the first stage, the formulation of a dynamic model of the wheelchair is carried out. The conventional differential drive structure of the wheelchair with two front castor wheels is considered (DeSantis, 2009; Mohareri et al., 2012). Its dynamic model derivation is based on the Euler Lagrange formalism (Uicker, 1969; Kahn &

Roth, 1971), and is carried out in two folds. Initially, both kinetic and gravitational energy is considered in the Lagrangian function to account for the wheelchair’s dy- namic properties on both inclined and non-inclined configurations without slipping situations. The slipping parameters are then formulated and incorporated into the model. The determination of the slipping parameters is approached from the geo- metric perspective, by considering the non-holonomic motions of the wheelchair in the Euclidean space. Because of its non-holonomic nature, the model constitutes the class of uncertain non-linear systems.

The study also involves the development of a steering behaviour model for wheelchair drivers. The formulation is based on the reactive potential field approach (Khatib, 1985; Koren & Borenstein, 1991; Jaradat et al., 2011), that has been considered

by numerous experimentally validated models in the literature (Jaradat et al., 2011).

The formulation and identification of the time-series empirical driver model is carried out on account of two fundamental sources of information comprising the general observation of wheelchair steering and the generated microscopic steering data. In particular, the directed potential field method is considered in the formulation of the driver’s risk detection and risk avoidance behaviours (Schneider & Wildermuth, 2005; Taychouri et al., 2007). The advantage of the proposed directed potential field method is that: apart from using the distance representation and taking the risks dissemination into account, it also allocates variable repulsive potential on the relative direction of the risk from the wheelchair. In the identification of the driver- specific steering behaviours, the ordinary least square procedure is considered in the computation of best-fitting driving model parameters.

At the final stage, the closed-loop model utilising the partial-state feedback con- troller is proposed in the tracking of user inputs by torque compensation (Codourey, 1998). The control of similar non-linear systems by feedback linearisation can either be full-state or partial-state. The full-state (or input-state) feedback linearisation involves complete linearisation of the system’s states with respect to control inputs by coordinate transformation and static state feedback, while the partial-state or the input-output feedback linearisation procedure linearises dynamics of the systems between the input and the output. In real-life however, the exact conditions for the full-state linearisation are only satisfied by few non-linear systems (Isidori, 1995; Hunt et al., 1983; Su, 1982). Because the proposed dynamic wheelchair model does not satisfy the full-state feedback linearisation conditions (Isidori, 1995), the partial-state feedback linearisation technique is considered. Nevertheless, the system is minimum- phase with stable internal dynamics. The optimality of the resulting closed-loop system is ensured through the performance index of the non-linear continuous-time generalised predictive controller (GPC).

### 1.5 Outline of the main contributions

The main contributions of this work can be summarised as follows:

• The identified driver-specific parameters of the driving behaviour model con- stitutes the equilibrium between the subjective risk level of the driver and the prevailing environmental situation. The formulation and implementation of a driving behaviour model using driver-specific parameters, to adapt the wheel- chair’s velocity to the driver’s behaviour to achieve the background steering assistance with minimum corrective adjustments on the steering signals, entail the main contributions of this thesis. Besides, the proposed assistive system employs the driver-specific parameters in the driving model to ensure both fine steering manoeuvres and automatic risk and collision avoidance behaviours.

• The study also contributes to the development of a wheelchair driving behaviour model that is simple and linear in the parameters, with a capacity to allocate directed reactive resources against sensor detectable risks. These attributes make the driving model implementable on-line as real-time intelligent co-driver, on board the wheelchair, that predicts and provides local corrective solutions to possible steering errors in accordance with the driver’s preference and current situation.

• The development of a dynamic model of a differential drive wheelchair and derivation of slipping parameters also constitutes a contribution of the study.

The dynamic model takes into account the effects of rolling friction, slipping parameter and gravitational potential of the wheelchair, on both inclined and non-inclined surfaces, and therefore presents a more realistic representation of the wheelchair.

• The study incorporates the driving behaviour model, the wheelchair model and a feedback controller in a closed-loop system, to adapt the control of the wheelchair to the driver’s behaviour to realise the intended background steering assistance.

### 1.6 Delineations and Limitations

This study is based on the following assumptions:

• In order to formulate the wheelchair dynamic model, it is considered that the wheelchair, including its components, is built from rigid bodies, and therefore possess no flexible links.

• It is presumed that the symmetric structure of wheelchair remains unchanged during use, implying that the centre of mass will always remain along the longitudinal axis of the wheelchair’s motion.

• In the modelling of slipping parameters, it is considered that the front cas- tor wheels are relatively far away from the centre of mass compared to the hind wheels, and therefore experience no longitudinal slip because of the re- duced force effect. The castor wheels velocity is thus presumed to represent the wheelchair’s absolute velocity.

• In the driving behaviour modelling, explicit knowledge of the driver’s subse- quent intentions is presumed to be available.

### 1.7 Publications

1. Onyango S.O., Hamam Y., Djouani K., Daachi B., & Steyn N. (2016). A Driving Behaviour Model of Electrical Wheelchair Users. Computational In- telligence and Neuroscience, 2016(2016), 20. http://doi.org/10.1155/2016/

7189267

2. Onyango S.O., Hamam Y., Djouani K., & Daachi B. (2016). Modeling a pow- ered wheelchair with slipping and gravitational disturbances on inclined and non-inclined surfaces. SIMULATION, 92(4), 337355. http://doi.org/10.1177/

0037549716638427

3. Onyango S.O., Hamam Y., Djouani K., & Daachi B. (2015). Identification of wheelchair user steering behaviour within indoor environments. In 2015 IEEE International Conference on Robotics and Biomimetics (ROBIO),. Zhuhai China: IEEE. http://ieeexplore.ieee.org/xpls/abs all.jsp?arnumber=7419114&

tag=1

### 1.8 Thesis Chapter Overview

Chapter 1 presents the general introduction as well as the background and motivation for the study. Chapter 2 reviews the previous and current insights about the practises and methodologies used in the modelling and control of systems with human-in-the- loop. This includes the current advancements in the modelling of the differential drive system, particularly, wheeled mobile robots and wheelchair systems. An overview of the existing user handling models is provided. Because of the structural similarities between cars and wheelchairs, the driving model developments for the two systems is also elaborated. Chapter 2 further includes a few methodologies used in the control of

non-linear systems. In Chapter 3, the proposed dynamic model of a differential drive wheelchair system is introduced. It presents the employed formulation and validation procedure of the wheelchair model as well as the derivation and incorporation of slip- ping parameters into the dynamic model. Chapter 4 presents the proposed wheelchair driving model. This also encompasses the formulation, parameter identification, and validation of the driver model. The closed-loop system with human-in-the-loop is presented in Chapter 5, while Chapter 6 provides the conclusion, and outlines a few opportunities for further research.

### CONTROL WITH HUMAN-IN-THE-LOOP METHODOLOGIES: A SURVEY

### 2.1 Introduction

This chapter presents the relevant previous and contemporary advancements in the control of dynamical systems with human-in-the-loop. It entails the existing mod- elling and identification contributions on both dynamical systems and operator be- haviours. The chapter elaborates the control methodologies employed to manage the integration of the dynamic wheelchair and the driving behaviour models in the con- trol loop. This includes the concept of time, kinematic structure and virtual work in system modelling, and analysis of microscopic user-data for system identification, which constitute the fundamental design requirements taken into consideration to evaluate the desirable operational behaviour of a system. Since the literature in this field is quite elaborate, a limited scope of the survey is necessitated. As a result, the chapter is limited to the modelling and control aspects that relate to wheeled-mobile system (WMS).

### 2.2 Background

A WMS is a mechanism with actuated and possibly non-actuated rolling wheels, mounted to provide both support and relative motion (Muir, 1988). The system not only consists of the main body and wheels, herein referred to as the moving platform, but also the surface upon which the platform moves. The application of WMSs is

15

quite ancient in the transportation sector, and more evident in the modern world, with continuing research and developments in the transportation, defence, medical and manufacturing sectors. The common structural configurations of the wheeled platforms include the differential drive with two non-steered driving wheels and one or more caster wheels for stability, the tricycle with two non-steered and one steered wheels, the synchro-structure, the steered Ackermann and the omnidirectional drive structure (Katevas, 2001). The differential drive structure produces a straight line motion when all its drive wheels are turned at the same rate in the same direction, and an in-place rotation with zero turning radius, given equal and opposite turning velocities. Owing to the simple configuration and easier odometry, the differential drive structures find diverse application in common user and robotic systems. This chapter focuses on the differential drive structure with two front or rear caster wheels as applied in the powered wheelchair (Ding & Cooper, 2005; DeSantis, 2009).

Based on the type and assembly of the driving wheels, holonomic and non-holonomic constraints may be imposed on the platform’s motion. In consequence, the motion a WMS and the implied complexity in its controllability is highly dependent on the structural configuration. The holonomic constraints relate the time and positional variables of a kinematic system. In the presence of holonomic constraints, the final state of a kinematic link, is only dependent upon the initial states of other con- nected links. This means that given the initial states, it is possible to compute both translational and rotational positions of the link from the linear and rotational po- sitions of the adjacent links. Besides, all velocity constraints can be integrated into positional constraints. As a result, all degrees of freedom (DoFs) related to a spa- cial kinematic system with holonomic constraints are easily controllable on planar surfaces with simple motion planning tasks (Mariappan et al., 2009). The omnidirec- tional holonomic wheels are numerous in the literature of mobile robotics. However, their main drawback entails the complexity of the wheeling system that necessitates

high energy besides the periodic maintenance required by the actuators (Xu, 2005;

El-Shenawy, 2010). Because of this, the holonomic wheels are rarely implemented in common daily applications. The configurations with non-holonomic constraints on the other hand, introduce a continuous closed-circuit of constraining parameters, that governs the transformations of the system from one state to the other (Bryant, 2006). Accordingly, the velocity constraints are non-integrable, indicating that the final state of the system depend on the transitional trajectory values within the pa- rameter space. The strength of the non-holonomic structures, nonetheless, lies in the construction simplicity, with fewer controllable axis required to ensure the necessary mobility. This makes them reliable, efficient, flexible and prevalent. Besides, the us- age of multiple disk shaped wheels improves the robustness and stability of systems with non-holonomic constraints in the presence of irregular terrains. However, non- holonomic systems are strongly non-linear, and require exhaustive non-linear analysis (Astolfi, 1996; Koon & Marsden, 1997). Thus, designing a good control system is generally a considerable challenge.

### 2.3 Modelling of WMSs

The majority of the literature concerns the kinematic and dynamic modelling. The kinematics of a WMS refers to the study of the system’s motion that results from the geometry of constraints of the wheels’ rotational motion (Muir, 1988). In kinematic modelling, the preceding requirement regards the allocation of numerous coordinate frames within the system and the environment, to facilitate the formulation of pa- rameters and variables of the kinematic model. The kinematic modelling parameters, include the angles and distances between the various coordinate systems, while the variables include the relative positions, velocities and accelerations of the body and

the wheels. Dynamic modelling on the other hand, determines the relationship be- tween the actuator forces and the resulting motion of the WMS. The parameters involved in dynamic modelling include the angles and distances between different coordinate systems, mass and inertia components, and frictional coefficients; while the variables include the positions, velocities and accelerations of the wheels and the body.

Notably, two approaches are available in the literature of kinematic and dynamic modelling of WMSs: the non-generic vector approach, based on geometric inter- pretation of global relationships between the centroid velocity and the joints’ rates (Kelly & Seegmiller, 2010; Byung-Ju Yi & Whee Kuk Kim, 2000); and the generic transformation approach that involves an outline of the system’s kinematic structure, with coordinate frames that may be assigned according to Sheth-Uicker’s convention (Sheth & Uicker, 1971; Muir, 1988; Holmberg & Khatib, 2000). In the transformation approach, the wheel Jacobian and joints transformation matrices are formulated to express the displacement relationships between the different links of a WMS.

### 2.3.1 Special kinematic characteristics of a WMS

The following special characteristics of a WMS also provide a distinction between the internal kinematics; that relates the different links of a mechanism, and the external kinematics; that provides a relationship between a mechanism and its environment (Schaal et al., 2003; Ambike & Schmiedeler, 2006).

1. Each wheel is in contact with the body and the surface of travel, forming as many parallel closed-chains as the number of wheels. As a result, WMSs neces- sitates parallel computation of both kinematic and dynamic models. Unlike the mobile systems, most stationary mechanisms (with exception of manipulators

whose end effectors are in contact with fixed objects) have open-chains with links that are serially connected by joints without closed-circuits. In conse- quence, the stationary mechanisms only require serial kinematic and dynamic modelling. According to Muir (1988), a mechanical structure that amounts to a closed-chain system can be represented by Figure 2.1. The structure, com- prising the main body, N open-chains, and the environment not only models WMSs, but also a variety of robotic mechanisms. Analogous to a WMS, the main body represents the body of the WMS, the N open-chains represent the N wheels, while the environment represents the surface of travel.

2

Open chains 4 … N

1 3

Environment Main body

Figure 2.1: A simple closed chain mechanism

2. A higher-pair pseudo joint, that enables rotational and translational motions with respect to the point of contact exists, between each wheel and the surface of travel. According to (Katevas, 2001), a pair is a joint between two bodies that keeps them not only in contact, but also in relative motion. A lower pair involves a surface contact, while a higher pair involves a point or a line contact.

Most stationary robotic mechanisms employ the lower-pair revolute, prismatic, helical, cylindrical, spherical or planar joints.

3. Unlike the open-chain mechanisms, where all joints must be actuated and sensed, it may be unnecessary to actuate and sense all the DoFs of the wheels to provide adequate control. Indeed, it is more favourable to compute the motion of non-actuated wheel because it is less likely to be affected by slippage.

4. Friction is important at the point of contact between the wheel and the surface of travel. The dry friction between the wheel and the surface of travel plays a very important role in ensuring the motion of the adjoining bodies. The friction in the wheel bearings is, however, undesirable because it results in excessive dissipation of energy.

### 2.3.2 Kinematic modelling of a differential drive system

By formulating the constraints that the joints impose on the adjacent links, the kine- matic models provide the basis for both dynamic modelling and model-based control.

In WMSs, the computed constraints include positional, velocity and acceleration con- straints of the body and the wheels, relative to the inertial coordinate system. This necessitates the simplifying assumption that the WMS is only built from rigid bod- ies, the transformation matrix and the wheel Jacobian matrix to describe and relate the translational and rotational motions associated with the joints. The relationship between the joints’ velocities may be computed by differentiating the corresponding positional relationships.

2.3.2.1 Coordinate system assignment

The conventional kinematic modelling procedure begins by assigning various coordi- nate frames to the various joints of a mechanism. Two conventions are commonly

applied in the assignment of coordinate frames: the Denavit-Hartenberg conven- tion (Niku, 2001), and the Sheth-Uicker convention (Sheth & Uicker, 1971). The Denavit-Hartenberg, also known as D-H convention, presents two displacements and two rotations characteristic parameters for attaching the coordinate frames to the links of a spacial kinematic chain. The convention entails a 4×4 homogeneous trans- formation matrix, that, apart from describing the size, the shape and the associated transformations of the link, also relates the successive coordinate frames on the kine- matic chain. Given the base effector’s coordinate vector, the transformation matrices may be cascaded from the base link to the end effector to determine the position and orientation of the end effector of a stationary robotic manipulator. This convention attaches one coordinate frame to every joint of the kinematic chain. In spite of its popularity, the D-H convention does not present an obvious joint ordering criteria, and therefore leads to ambiguous transformation matrices, especially in systems with multiple closed-chains like WMS where one link, the environment, associates more than two joints (Katevas, 2001). Sheth-Uicher convention solves this problem by assigning one coordinate frame at the end of each link, implying that each joint will have two coordinate axis (Muir, 1988). In a WMS, the links include the surface of travel and the body, while the joints connecting the two links are the wheels and the center of mass of the WMS. The latter joint is not physical, but rather a relationship between the body and the surface of motion.

In order to formulate the kinematic model, a stationary or inertial coordinate frame may be assigned on the surface of travel to provide an absolute reference for the sys- tem’s motion. The motion of a coordinate frame fixed at a point in the body relative to the inertial coordinate frame, herein referred to as body-fixed coordinate frame, may be interpreted as the WMS’s motion. It is noted, that although the choice of po- sition of origin and orientation of coordinate frames is not unique, it is preferred that positions and orientations which produce the appropriate formulation of a kinematic

model is considered. Depending on the number of wheels, the coordinate frames may be assigned at the point of contact between the surface of travel and the wheel. Each wheel and the main body can then be modelled as a planer pair with two or more DoFs, contingent on the associated kinematic constraints.

2.3.2.2 The homogeneous transformation matrix

Spacial kinematics may be regarded as a way of representing the rigid body’s pose and
displacement (translational and/or rotational motions) within a space. The 4×4 ho-
mogeneous transformation matrix consolidates both positional vector and rotational
displacement matrix in a compact matrix notation. The matrix is used in kinematic
modelling to transform a point’s coordinate to its corresponding coordinate in an-
other coordinate frame, such that, given the position of origin of frameAwith respect
to frame B, denoted by ^{B}r_{A}=_{B}

r^{x}_{A} ^{B}r^{y}_{A} ^{B}r^{z}_{A}T

, and the corresponding orientation,
computed using the rotation matrix of direction cosines, ^{B}R_{A}, in Equation (2.3), any
position vector ^{A}rin frameA can be transformed into position vector^{B}r in frameB
by expression (2.1), or expression (2.2) in matrix form.

Br=^{B} R_{A} ^{A}r+^{B}r_{A} (2.1)

Br 1

=

BR_{A} ^{B}r_{A}
0^{T} 1

Ar 1

(2.2)

where

BRA=

(¯xA·x¯B) (¯yA·x¯B) (¯zA·x¯B)
(¯x_{A}·y¯_{B}) (¯y_{A}·y¯_{B}) (¯z_{A}·y¯_{B})
(¯x_{A}·z¯_{B}) (¯y_{A}·z¯_{B}) (¯z_{A}·z¯_{B})

, (2.3)

with (¯xA y¯A z¯A) and (¯xB y¯B z¯B) representing the unit basis vectors of the frames
A andB respectively. The component,^{B}T_{A} =

"

BR_{A} ^{B}r_{A}
0^{T} 1

#

in Equation (2.2) and
(2.4), is the 4×4 homogeneous transformation matrix, consisting of four sub-matrices
namely: the rotation matrix, the position vector, the perspective transformation and
the scaling. Any transformation matrix ^{C}TA=^{C} TB BTA, may be computed with a
strict consideration of the transformation order.

BTA=

BR_{A(3×3)} ^{B}r_{A(3×1)}
0^{T}(1×3) 1(1×1)

=

rotation matrix position vector perspective vector scalar

(2.4)

In WMSs, the relative change in position and orientation of the body with respect to the surface of motion result from the wheels’ rotational motion. The wheel Jacobian matrix is used in kinematic modelling to relate the rotational motion of the wheels to the body’s motion. The analysis of Jacobian matrix has been considered the main tool for evaluating the kinematic performance of robotic manipulators (Tarokh

& McDermott, 2005; Galicki, 2016; Kanzawa et al., 2016). Several guidelines, in- cluding manipulability, condition number, isotropy and global conditioning index for kinematic performance, have been proposed (Merlet, 2007). An isotropic Jacobian matrix is emphasised because it establishes a linear map between the joints’ and the body’s velocities, ensuring that each actuator is providing a proportional effort in the body’s direction of motion (Zaw, 2003; Singh & Santhakumar, 2016). According to Muir (1988) and Ostrovskaya (2000), derivation of wheel Jacobian matrix is based directly on the velocity transformation matrices.